September 09, 2009
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# Introduction to Mesoscale Methods

Lecture at the Understanding Molecular Simulations school at the Jawaharlal Nehru Center for Advanced Scientific Research.

September 09, 2009

## Transcript

Chennai
2. ### Outline • What is mesoscale ? • Mesoscale statics and

dynamics through coarse-graining. • Coarse-grained equations for a binary ﬂuid-ﬂuid mixture. • Numerical methods of solution. • Example simulations.
3. ### Numbers and methods 1 2 3 4 5 6 7

0 Ab initio methods Atomistic methods Continuum methods DFT CPMD MD LD BD DPD CFD LB Number of atoms (log) Simulation method Example of method
4. ### How lengths scale with numbers !! Try this for water

Molar mass = 18 gm Molar volume = 18 mL N = 1023 L 107˚ A L N 1 3 N = 1 L 1˚ A !! Are other scalings possible ? How does length grow with size for a polymer ? Throughout, (!!) indicate exercises/derivations/points to ponder.
5. ### Length and time scales pico milli micro nano nano micro

milli L T H = E m ¨ R = ⇥U ˙ R = U ˙ c = D 2c !! Are there systems with millimeter L but picosecond T ?
6. ### How times scale with masses and lengths m¨ x +

kx = 0 2 0 = k m 0 m k ¨ u = c2 2u 0 = ±cq ⇥0 c ˙ u = D 2u i 0 = Dq2 ⇥0 2 D Harmonic Oscillator Wave Equation Diffusion Equation
7. ### Thermal ﬂuctuations P(x |x) = 1 ⇤ 2 D⇥ exp

(x x)2 2D⇥ ⇥ Brownian motion : Einstein (1905) D = kBT 6⇥ a temperature size Stokes-Einstein-Sutherland Relation Q. Why does a smaller particle have a higher diffusion coefﬁcient ? A. The ‘root N’ effect! The central limit theorem helps us understand this. !! Is it (central limit) theorem or central (limit theorem) ?
8. ### Langevin theory inertia damping reversible force ﬂuctuation deterministic stochastic mean

behaviour ‘root N’ ﬂuctuations regression to the mean equilibrium ﬂuctuations v2⇥ = kBT F = 0 Free Brownian particle m˙ v + v F(x) = ⇥(t)
9. ### Mesoscale regime 1 2 3 4 5 6 7 0

Ab initio Atomistic Continuum Number of atoms Simulation Coarse grained length scales. Coarse grained time scales. Retain thermal ﬂuctuations. Mesoscale methods } Examples Brownian dynamics. Dissipative particle dynamics. Time-dependent Ginzburg-Landau.
10. ### Coarse-graining in degrees of freedom What is the idea ?

x1 x2 } P(x1, x2 ) RV y = x1 + x2 “Coarse-grained” sum P(y) P(y) = dx1dx2 (y x1 x2 )P(x1, x2 ) What is the distribution ? contains less information about the system than P(y) P(x1, x2 ) Adequate if we are only interested in the sum variable.
11. ### General ‘coarse-graining’ formula P(x1, x2, . . . , xN

) y = f(x1, x2, . . . , xN ) Microstate probability Mesoscale variable/order parameter P(y) = ⇥ N i=1 dxi [y f(x1, . . . , xN )]P(x1, . . . , xN ) Mesostate probability
12. ### Coarse-grained Landau-Ginzburg functional Microscopic Hamiltonian Gibbs distribution P(q) = exp[

H(q)] Z H(q) = f(q) Order parameter Deﬁnition of Landau functional P(⇤) = ⇥ N i=1 dqi⇥[⇤ f(q)] exp[ H(q) Z ⇥ exp[ L] Z P(⇥) ⇥ exp[ L] Z
13. ### Explicit DOF coarse-graining ... H(s1, s2, . . . ,

sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si
14. ### Explicit DOF coarse-graining ... H(s1, s2, . . . ,

sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si P(⇥ ) ⇥ exp[ L] Z⇥ =? K. Binder, Z. Phys B, 43, 119 (1981)
15. ### Explicit DOF coarse-graining ... H(s1, s2, . . . ,

sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si P(⇥ ) ⇥ exp[ L] Z⇥ =? K. Binder, Z. Phys B, 43, 119 (1981)
16. ### Explicit DOF coarse-graining ... H(s1, s2, . . . ,

sN ) = J ij⇥ sisj Order parameter. Block spin. Coarse-grained magnetization. = 1 N i si P(⇥ ) ⇥ exp[ L] Z⇥ =? K. Binder, Z. Phys B, 43, 119 (1981) Explicit DOF needs this!
17. ### Or ansatz based on symmetry. L = A 2 2

+ B 4 4 + K 2 (⇥ )2 P(⇥) ⇥ exp[ (A 2 ⇥2 + B 4 ⇥4)] P(s) ⇥ a+ (s 1) + a (s + 1) -J +J local part non-local part K( )2 All equal-time equilibrium properties are determined by this functional. !! Why is this called a “free energy” ? Is it a thermodynamic free energy ?
18. ### What about dynamics ? Temporal coarse-graining D = kBT 6⇥

a Why is this independent of particle mass ? ⇥d = m Inertial time scale. Time the velocity ‘remembers’ its initial condition. d ˙ x = v ˙ x = ⇥(t) Overdamped equations of motion. Valid when d m˙ v + v = ⇤(t) = 6⌅⇥a
19. ### Overdamped stochastic equations damping reversible force ﬂuctuation deterministic stochastic mean

behaviour ‘root N’ ﬂuctuations ˙ x = ⇥U + ⇥(t)
20. ### Overdamped coarse-grained equations of motion ˙ ⇤(r) = L ⇤

+ ⇥(r, t) ˙ ⇥(r) = [A⇥ + B⇥3 K⇥2⇥] + (r, t) damping reversible force ﬂuctuation Model A equation ⇥⇥(r, t)⇥(r , t )⇤ = 2kBT (r r ) (t t ) Fluctuation Dissipation Relation. !! Zero-mean Gaussian noise (r, t)⇥ = 0 “Stochastic partial differential equations with additive Gaussian noise”
21. ### Conserved order parameters Ising spin Lattice gas Ising spins are

usually non-conserved d dt (r, t) = 0 Lattice gas models are conserved d dt (r, t) = 0
22. ### Conserved coarse-grained dynamics damping reversible force ﬂuctuation Model B equation

Fluctuation Dissipation Relation. !! Zero-mean vector Gaussian noise (r, t)⇥ = 0 ˙ ⇤(r) = ⇤2 L ⇤ + ⇤ · ⇥(r, t) ˙ ⇥(r) = ⇤2[A⇥ + B⇥3 K⇤2⇥] + ⇤ · (r, t) !! Derive this. Use the local conservation law ˙ + ⇥ · j = 0 ⇥⇥i (r, t)⇥j (r , t )⇤ = 2kBT ij (r r ) (t t )
23. ### Numerical solution Stochastic PDE Stochastic ODE Stochastic realisation Spatial discretisation

Temporal integration ⇤t⇥ = D⇤2 x ⇥ + ⇥x = (x + h) (x) h ⇤t⇥(i) = DL2 ij ⇥(j) + (i) ⇥(i, t + t) ⇥(i, t) = t+ t DL2 ij ⇥(j) + (i) !! Obtain the explicit form of the L matrix for a central difference Laplacian
24. ### Summary of TDGL mesoscale methods Microscopic Hamiltonian Ginzburg-Landau Functional Langevin

equations Explicit Coarse Graining Overamped approximation Symmetry + gradient expansion Numerical solution
25. ### Conclusion • Mesoscale methods are appropriate at length scales intermediate

between the molecular and continuum scales. • The continuum description must be supplemented by ﬂuctuation terms. At the mesoscale, the number of particles is not so large that ﬂuctuations can be neglected. • Lengths and times must be coarse-grained. Intelligent coarse-graining improves computational eﬃciency, often by orders of magnitude, compared to direct MD. • Mesoscale methods can be particle-based, for instance dissipative particle dynamics and Brownian dynamics, or ﬁeld-based like time-dependent Ginzburg-Landau theory and ﬂuctuating hydrodynamics. • TDGL equations can be solved very eﬃciently on computers using matrix formulations.
26. ### Further reading • Stochastic processes in Physics and Chemistry :

van Kampen • Handbook of stochastic processes : Gardiner • Principles of Condensed Matter Physics : Chaikin and Lubensky • Modern Theory of Critical Phenomena : Ma • Reviews of Modern Physics article : Halperin and Hohenberg • Numerical Solutions of Stochastic Diﬀerential Equations : Kloeden and Platen.